The gluon beam function at two loops

The virtuality-dependent beam function is a universal ingredient in the resummation for observables probing the virtuality of incoming partons, including N -jettiness and beam thrust. We compute the gluon beam function at two-loop order. Together with our previous results for the two-loop quark beam function, this completes the full set of virtuality-dependent beam functions at next-to-next-to-leading order (NNLO). Our results are required to account for all collinear initial-state radiation effects on the N -jettiness event shape through N3LL order. We present numerical results for both the quark and gluon beam functions up to NNLO and N3LL order. Numerically, the NNLO matching corrections are important. They reduce the residual matching scale dependence in the resummed beam function by about a factor of two.